There have been many threads here about expensive audio gear in the context of relationships. Unfortunately they tend to degenerate into questionable advice and stereotypes.

First off, you shouldn't assume she's incapable of ever understanding. If you begin with this assumption communication is going to be a non option from the get go, and communication is an important facet of a successful relationship. I've spoken with audiophiles and have read plenty of comments in the past where the situation is treaded like a game of deception and one-upsmanship, basically expending time and effort to pull the wool over their eyes. In these cases the partner in question often has a conniption fit, and it has less to do with money in and of itself and more to do with the fact that a large purchase wasn't discussed before hand.

Let's be honest here: a 1000 USD pair of headphones is pretty unusual by most peoples' standards. A 500 USD pair of headphones is pretty unusual by most peoples' standards.

Maybe you can convince her it's worth its steep price tag as a piece of equipment. Maybe you can't. However you may very well have more success in convincing her it's part of your hobby and something from which you derive a lot of enjoyment. Try to explain to her why you enjoy these types of things. Explain to her that you hear a difference between one headphone and another. Even if she doesn't feel the same way or "get it" from a hobbyist's standpoint, she may feel more at ease if you involve her in the process. Suggest that you two listen together to her favorite music once you get the new headphones. Just keep in mind that this stuff takes patience: you shouldn't very well expect to throw headphones on someone with no experience and have them appreciate the improvements in sound right away.

Even if she can't hear what all the fuss is about, she should at least enjoy spending the time with you and being involved in your interests. In that sense you're not convincing her of the value of headphones, but rather the value of your headphone hobby. Establishing boundaries is crucial, and as an adult she should realize you have your own interests as she has hers. Even so, keeping the channel of communication open is important,

Even if she can't hear what all the fuss is about, she should at least enjoy spending the time with you and being involved in your interests. In that sense you're not convincing her of the value of headphones, but rather the value of your headphone hobby.

Take this advice. It's not the headphones or an amp or a source. It's the hobby. Don't try to hide it, just talk to her about it.

Make sure you're not mortgaging anything (house, kids' college education) to supply your habit, though. That's a different conversation.

Does she have any intense passions of her own? Try to draw parallels to those passions.

There are extremes to every hobby and pastime; some people are really into numismatics and collect every manner of coin out there; then again, change is in your pockets every single day and most people never give two thoughts about whether that 1913 Wheat Penny is worth anything. Your interest just happens to be in a very utilitarian field of acoustics that most people think can be done perfectly fine with $5 earbuds.

If you can draw parallels about the passion and precision care used by people like Audeze to craft the LCD-2 into a music-making machine and the way something else (that she thinks is more of a worthy endeavor) holds the same spirit of passion and care, then you might be able to appeal to her senses.

Lastly, she's your spouse, and the likely assumption is that if you truly have a deep passion for listening to music in as high-quality a manner as possible, then she can and should be able to identify it in you. If you show, with your words and your actions, that you care deeply about musical reproduction, then she'll at least understand that this is a true interest of yours. If you're not able to show that fact convincingly in yourself, then perhaps it's best to first examine your own reasons for wanting to purchase a high-end headphone.

Headphone audio is less expensive a hobby than golf, cars or even woodworking. $4000 will buy a top-notch system that will serve you for years but that wouldn't begin to subsidize a car habit or pay for a club golf membership and green fees for a year, and don't even talk about woodworking, my friend has well over $10,000 invested in tools. Make the point it's not that expensive a hobby in comparison to others. Good luck.

Headphone audio is less expensive a hobby than golf, cars or even woodworking. $4000 will buy a top-notch system that will serve you for years but that wouldn't begin to subsidize a car habit or pay for a club golf membership and green fees for a year, and don't even talk about woodworking, my friend has well over $10,000 invested in tools. Make the point it's not that expensive a hobby in comparison to others. Good luck.

Problem here is that some people exchange greens fees for their expanding gear collection. Never satisfied, if you know what I mean.

In general, its hard to justify a hobby to someone else who doesn't share the same interest. However, it should be a hobby that doesn't demand this kind of expenditure all the time. In other words, be smart with your choices, now that you're going to buy a 1k headphone, don't go around spending on 5k amps and dacs that don't really matter (unless you're a millionaire of course).

We don't know about your situation money wise, so keep that in mind as well.